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Chapter 11: Problem 76

Use the summation properties and rules to evaluate each series. $$\sum_{i=1}^{6}\left(i^{2}+2 i^{3}\right)$$

Short Answer

Expert verified
The evaluated series is 973.

Step by step solution

01

Separate the Summation

Use the summation property \( \sum (a_i + b_i) = \sum a_i + \sum b_i \) which allows breaking the sums into separate parts: \[ \sum_{i=1}^{6}(i^2 + 2i^3) = \sum_{i=1}^{6} i^2 + \sum_{i=1}^{6} 2i^3 \]
02

Factor Out Constants

Factor out any constants from the summation: \[ \sum_{i=1}^{6} i^2 + \sum_{i=1}^{6} 2i^3 = \sum_{i=1}^{6} i^2 + 2\sum_{i=1}^{6} i^3 \]
03

Use Summation Formulas

Utilize known summation formulas: \ For the sum of squares: \ \(\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6}\) \ For the sum of cubes: \ \(\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2\) \ Plugging in \(n = 6\): \ \(\sum_{i=1}^{6} i^2 = \frac{6(6+1)(2\cdot6+1)}{6} = \frac{6\cdot7\cdot13}{6} = 91\) \ \(\sum_{i=1}^{6} i^3 = \left(\frac{6(6+1)}{2}\right)^2 = \left(\frac{6\cdot7}{2}\right)^2 = 21^2 = 441\)
04

Compute the Final Sum

Combine the results from Step 3 to compute the final summation: \[ \sum_{i=1}^{6} i^2 + 2\sum_{i=1}^{6} i^3 = 91 + 2\cdot 441 = 91 + 882 = 973 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Sum of Squares
The 'sum of squares' refers to the addition of squared numbers in a series. It's common in algebra and is crucial for understanding patterns in sequences and series.
To make calculations easier, we use the summation formula for squares:

\[\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \]
This formula helps quickly compute the sum of squares for any number of terms without manually squaring and adding each number.
Sum of Cubes
The 'sum of cubes' involves the addition of cubed numbers in a series. It's especially useful in more advanced algebraic problems. The summation formula for cubes is:
\[\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2 \]
This formula shows that the sum of cubes up to a certain number n is the square of the sum of the first n natural numbers. Knowing this formula simplifies problems significantly and avoids lengthy calculations of individual cubes.
Summation Formulas
Summation formulas are essential tools in algebra. They allow us to evaluate series quickly and efficiently. These formulas take the tedious work out of summing individual terms.
Some key summation formulas include:
  • Sum of natural numbers: \[\sum_{i=1}^{n} i = \frac{n(n+1)}{2} \]
  • Sum of squares: \[\sum_{i=1}^{n} i^2 = \frac{n(n+1)(2n+1)}{6} \]
  • Sum of cubes: \[\sum_{i=1}^{n} i^3 = \left(\frac{n(n+1)}{2}\right)^2 \]

These formulas make it easier to handle large datasets or complex algebraic questions. For example, in the original exercise, using these formulas simplified the problem and led directly to the correct answer.

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Most popular questions from this chapter

The code for some garage door openers consists of 12 electrical switches that can be set to either 0 or 1 by the owner. With this type of opener, how many codes are possible? (Source: Promax.)

Assume that \(n\) is a positive integer. Use mathematical induction to prove each statement S by following these steps. See Example \(I\). (a) Verify the statement for \(n=1\) (b) Write the statement for \(n=k\) (c) Write the statement for \(n=k+1\) (d) Assume the statement is true for \(n=k\). Use algebra to change the statement in part (b) to the statement in part (c). (e) Write a conclusion based on Steps (a)-(d). $$\frac{1}{1 \cdot 2}+\frac{1}{2 \cdot 3}+\frac{1}{3 \cdot 4}+\dots+\frac{1}{n(n+1)}=\frac{n}{n+1}$$

If a bag contains 15 marbles, how many samples of 2 marbles can be drawn from it? How many samples of 4 marbles can be drawn?

The table gives the results of a 2008 survey of Americans aged \(18-24\) in which the respondents were asked, "During the past 30 days, for about how many days have you felt that you did not get enough sleep?"$$\begin{array}{|l|c|c|c|c|}\hline \text { Number of Days } & 0 & 1-13 & 14-29 & 30 \\ \hline \text { Percent (as a decimal) } & 0.23 & 0.45 & 0.20 & 0.12 \\\\\hline\end{array}$$ Using the percents as probabilities, find the probability that, out of 10 respondents in the \(18-24\) age group selected at random, the following were true. Fewer than 2 did not get enough sleep on 14 or more days.

Find all natural number values for \(n\) for which the given statement is false. $$2^{n}>n^{2}$$
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